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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Sharp Jackson-type inequalities for approximations of classes of convolutions by entire functions of exponential type


Author: O. L. Vinogradov
Translated by: the author
Original publication: Algebra i Analiz, tom 17 (2005), nomer 4.
Journal: St. Petersburg Math. J. 17 (2006), 593-633
MSC (2000): Primary 41A17
Published electronically: May 3, 2006
MathSciNet review: 2173937
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, a new method is introduced for the proof of sharp Jackson-type inequalities for approximation of convolution classes of functions defined on the real line. These classes are approximated by linear operators with values in sets of entire functions of exponential type. In particular, a sharp Jackson-type inequality for the even-order derivatives of the conjugate function is proved. For the uniform and the integral norm, the estimates are sharp even if their left-hand sides are replaced by the best approximation. Sharp inequalities for approximations of periodic functions by trigonometric polynomials and of almost-periodic functions by generalized trigonometric polynomials are special cases of the inequalities mentioned above.


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Additional Information

O. L. Vinogradov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ Prospect 28, Staryĭ Peterhof, St. Petersburg 198904, Russia
Email: olvin@math.spb.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-06-00922-8
PII: S 1061-0022(06)00922-8
Keywords: Jackson inequalities, sharp constants, entire functions of exponential type
Received by editor(s): November 30, 2004
Published electronically: May 3, 2006
Additional Notes: Supported by the “Universities of Russia” program (project no. ur.04.01.036) and the “Leading scientific schools” program (project no. NSh-2266.2003.1)
Article copyright: © Copyright 2006 American Mathematical Society